Proof: Do Control Map At Unchanged 44

Let's prove the following theorem:

if the following are true:

expression state at time 44 = "call_function_begin"
Control Map at time 44 = [ entry 0: (pair ("while", 2)), [ ] ]

then Control Map at time 45 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof:

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Given

1	expression state at time 44 = "call_function_begin"
2	Control Map at time 44 = [ entry 0: (pair ("while", 2)), [ ] ]

Proof Table

#	Claim	Reason
1	Control Map at time (44 + 1) = Control Map at time 44	if expression state at time 44 = "call_function_begin", then Control Map at time (44 + 1) = Control Map at time 44 (Assign Parent (9))
2	Control Map at time (44 + 1) = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (44 + 1) = Control Map at time 44 and Control Map at time 44 = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time (44 + 1) = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality)
3	44 + 1 = 45	44 + 1 = 45 (Fixed Value Statement)
4	Control Map at time (44 + 1) = Control Map at time 45	if 44 + 1 = 45, then Control Map at time (44 + 1) = Control Map at time 45 (Substitution)
5	Control Map at time 45 = [ entry 0: (pair ("while", 2)), [ ] ]	if Control Map at time (44 + 1) = Control Map at time 45 and Control Map at time (44 + 1) = [ entry 0: (pair ("while", 2)), [ ] ], then Control Map at time 45 = [ entry 0: (pair ("while", 2)), [ ] ] (Transitive Property of Equality Variation 2)

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